The map has dark and light areas that are perfect squares (tiles). What you want to do is ride your bike around the entire map so that there are no dark areas left. If you've been playing a while you might have to go to some out of the way places to see the dark areas, check the edges of the cliffs and the far ends of the beaches and you'll see it more clearly.

Hello clrl,
Additional to Nienna's posting look in the Wii Fit Forum Wiki:
http://www.wiifitforum.org/wiki/index.php/Island_Cycling – "The gift bike"

Also, it's only on the Free Cycle course--not any of the others.

The tiles light up starting from the Beginners course.
Starting from now, yes.
Look at the first tile at the first start and you can also see the size (all other equal sized) of one tile on this picture.

They do? I never ever noticed any shading until I started on the Free Cycle course.

I also noticed it first time in a later course, but I checked it yesterday with a brand new Mii and saw it right from the very begin.

The map is divided into squares that will lighten if you've gone through them at some time. You need to hit all the dark ones and make them turn light.

Thank you. I found the tiles and after two runs (1 hour) I completed all the island according to a message that flashed on my screen.
I still haven't got a new bike. what else is needed? I understand that doing all the balloons doesn't help. Do I have to get enough distance in 30 min. for 4 starts?

Only a new start of the game.

Got a new Bike!!
tnks